Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms

We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions--the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for constructing 2D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1D counterpart, we relate the real and imaginary components of these complex wavelets using a multi-dimensional extension of the HT--the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient FFT-based filterbank algorithm for implementing the associated complex wavelet transform.Comment: 36 pages, 8 figure

Brain Dynamics across levels of Organization

After presenting evidence that the electrical activity recorded from the brain surface can reflect metastable state transitions of neuronal configurations at the mesoscopic level, I will suggest that their patterns may correspond to the distinctive spatio-temporal activity in the Dynamic Core (DC) and the Global Neuronal Workspace (GNW), respectively, in the models of the Edelman group on the one hand, and of Dehaene-Changeux, on the other. In both cases, the recursively reentrant activity flow in intra-cortical and cortical-subcortical neuron loops plays an essential and distinct role. Reasons will be given for viewing the temporal characteristics of this activity flow as signature of Self-Organized Criticality (SOC), notably in reference to the dynamics of neuronal avalanches. This point of view enables the use of statistical Physics approaches for exploring phase transitions, scaling and universality properties of DC and GNW, with relevance to the macroscopic electrical activity in EEG and EMG

Numerical investigations of discrete scale invariance in fractals and multifractal measures

Fractals and multifractals and their associated scaling laws provide a quantification of the complexity of a variety of scale invariant complex systems. Here, we focus on lattice multifractals which exhibit complex exponents associated with observable log-periodicity. We perform detailed numerical analyses of lattice multifractals and explain the origin of three different scaling regions found in the moments. A novel numerical approach is proposed to extract the log-frequencies. In the non-lattice case, there is no visible log-periodicity, {\em{i.e.}}, no preferred scaling ratio since the set of complex exponents spread irregularly within the complex plane. A non-lattice multifractal can be approximated by a sequence of lattice multifractals so that the sets of complex exponents of the lattice sequence converge to the set of complex exponents of the non-lattice one. An algorithm for the construction of the lattice sequence is proposed explicitly.Comment: 31 Elsart pages including 12 eps figure

Non-Gaussian statistics in space plasma turbulence, fractal properties and pitfalls

Magnetic field fluctuations in the vicinity of the Earth's bow shock have been investigated with the aim to characterize the intermittent behaviour of strong plasma turbulence. The observed small-scale intermittency may be the signature of a multifractal process but a deeper inspection reveals caveats in such an interpretation. Several effects, including the anisotropy of the wavefield, the violation of the Taylor hypothesis and the occasional occurrence of coherent wave packets, strongly affect the higher order statistical properties. After correcting these effects, a more Gaussian and scale-invariant wavefield is recovered.Comment: 13 pages (including 13 postscript figures), to appear in Nonlinear Processes in Geophysic